Deterministic Brownian motion generated from Brownian motion generated from differential delay...

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  • PHYSICAL REVIEW E 84, 041105 (2011)

    Deterministic Brownian motion generated from differential delay equations

    Jinzhi LeiZhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, China

    Michael C. MackeyDepartments of Physiology, Physics, and Mathematics, and Centre for Applied Mathematics in Bioscience and Medicine (CAMBAM),

    McGill University, 3655 Promenade Sir William Osler, Montreal, Quebec, Canada H3G 1Y6(Received 9 May 2011; published 6 October 2011)

    This paper addresses the question of how Brownian-like motion can arise from the solution of a deterministicdifferential delay equation. To study this we analytically study the bifurcation properties of an apparently simpledifferential delay equation and then numerically investigate the probabilistic properties of chaotic solutions ofthe same equation. Our results show that solutions of the deterministic equation with randomly selected initialconditions display a Gaussian-like density for long time, but the densities are supported on an interval of finitemeasure. Using these chaotic solutions as velocities, we are able to produce Brownian-like motions, which showstatistical properties akin to those of a classical Brownian motion over both short and long time scales. Severalconjectures are formulated for the probabilistic properties of the solution of the differential delay equation.Numerical studies suggest that these conjectures could be universal for similar types of chaotic dynamics,but we have been unable to prove this.

    DOI: 10.1103/PhysRevE.84.041105 PACS number(s): 05.40.Ca, 05.40.Jc, 05.45.Ac


    In 1828, Robert Brown reported his observations of theapparently erratic and unpredictable movement of smallparticles suspended in water, a phenomena now knownas Brownian motion. Almost three-quarters of a centurylater, a theoretical (and essentially molecular) explanation ofthis macroscopic motion was given by Einstein, in whichBrownian motion is attributed to the summated effect ofa vary large number of tiny impulsive forces delivered tothe macroscopic particle being observed [1] (A nice Englishtranslation of this and other works of Einstein on Brownianmotion can be found in Furth [2]). Brownian motion hasplayed a central role in the modeling of many randombehaviors in nature and in stochastic analysis and formedthe basis for the development of an enormous branch ofmathematics centered around the theory of Wiener pro-cesses.

    Since Brownian motion is typically explained as thesummated effect of many tiny random impulsive forces, it is ofinterest to know if and when Brownian motion can be producedfrom a deterministic process (also termed as deterministicBrownian motion) without introducing the assumptions typ-ically associated with the theory of random processes. Studiesstarting from this premise have been published in the pastseveral decades, and there are numerous investigations thathave documented the existence of Brownian-like motion fromdeterministic dynamics in both discrete time maps and flows[39]. These models have included the motion of a particlesubjected to a deterministic but chaotic force (also knownas microscopic chaos) [3,5] or a many-degree-of-freedomHamiltonian [8,9]. Experimental evidence for deterministicmicroscopic chaos was reported in Ref. [6] by the observationof Brownian motion of a colloidal particle suspended in water(cf. Ref. [10] for a more tempered interpretation, and Ref. [11](chap. 18) for other possible interpretations of experimentslike these).

    Several investigators have shown that a Brownian-likemotion can arise when a particle is subjected to impulsivekicks, whose dynamics are modeled by the following equations[3,12,13] {


    = vmdv

    dt= v + f (t).


    In Eq. (1), f is taken to be a fluctuating force consisting ofa sequence of -function-like impulses given by, for example,

    f (t) = m

    n=0 (t)(t n ), (2)

    and is a highly chaotic deterministic variable generated by (t + ) = T ( (t)), where T is an exact map or semidynamicalsystem, e.g., the tent map on [1,1] (for more discussionsand terminologies, see Refs. [7,13] and references therein).In the Eqs. (1) and (2), the impulsive forces are describedby (t)(t n ), which are assumed to be instantaneouslyeffective and independent of the velocity v(t). Dynamicalsystems of the form (1) have received extensive attentionand are known to be able to generate a Gaussian diffusionprocess [3,1215].

    In this study, we sought an alternative continuous timedescription of the random force f (t), which was assumed todepend on the state (velocity) of a particle, but with a lag time , i.e.,

    f (t) = F (v(t )), (3)and where F has the appropriate properties to generate chaoticsolutions. Thus, we consider the following differential delayequation {


    = vmdv

    dt= v + F (v(t )),

    v(t) = (t), t 0,(4)

    041105-11539-3755/2011/84(4)/041105(14) 2011 American Physical Society


    where (t) denotes the initial function that must alwaysbe specified for a differential delay equation. The secondequation in Eq. (4) is known to have chaotic solutions forsome forms of the nonlinear function F ; for example, seeRefs. [1620]. In these cases, the force F (v(t )) is certainlydeterministic but also unpredictable (in practice but not inprinciple) given knowledge of the initial function. In thispaper, we will examine how a Brownian motion can beproduced by the differential delay equation (4). In particular,we investigate the statistical properties of the velocity v(t),and [x(t)]2, the mean square displacement (MSD), of thesolutions defined by Eq. (4). We note that unlike Eq. (1),which is linear and nonautonomous, Eq. (4) is a nonlinearautonomous system. Numerical simulations have shown thatthe second equation in Eq. (4) can generate processes with aGaussian-like distribution [16,18]. Nevertheless, to the best ofour knowledge, there is no analytic proof for the existence ofBrownian motion based on the differential delay equation (4).

    We first make some observations about the second equationin Eq. (4) that determines the dynamics of the velocity. Asimple form of the random force is binary and fluctuatesbetween f0, for instance, given by

    F (v) = 2f0{H (sin(2v)) 12

    }, (5)

    where H is the Heavyside step function, i.e.,

    H (v) ={

    0 for v < 0

    1 for v 0. (6)

    We then have following equation


    dt= v + 2

    {H (sin(2v(t 1))) 1



    (7)v(t) = (t), t 0.

    Here and later we always assume the mass m = 1 and f0 = 1that can be achieved through the appropriate scaling. Thedelay differential equation (7) with a binary random forcecan be solved iteratively by the method of steps.1 Despite itssimplicity, it can display behaviors similar to a random process.An example solution of Eq. (7) is shown in Fig. 1, which lookslike noise.

    The random force in Eq. (7) is discontinuous and givesa continuous zigzag velocity curve (cf. inset in Fig. 1). In

    1A solution of Eq. (7) is associated with a time sequence t0 < t1 < < tn < , which is defined such that sin(2v(t)) 0 when t [t2k,t2k+1) and sin(2v(t)) < 0 when t [t2k1,t2k). Furthermore,if the sequence (t0, . . . ,tn) is known, then the solution v(t) whent (tn,tn + 1) can be obtained explicitly, and therefore, tn+1, whichis defined as sin(2v(tn+1)) = 0, is determined by (t0, . . . ,tn). Oncewe obtain the entire sequence {tn}, the solution of Eq. (7) consistsof exponentially increasing or decreasing segments on each interval[tn,tn+1]. Nevertheless, the nature and properties of the map tn+1 =Fn(t0,t1, . . . ,tn) is still not characterized and has defied analysis todate.

    0 20 40 60 80 1000.5








    FIG. 1. (Color online) A sample solution of (7) with = 10, = 1, and an initial function (t) 0.1,t [1,0]. The rectangu-lar inset shows the solution segment for 98 t 100.

    this paper, we will instead study an analogous but differentdifferential delay equation


    dt= v + sin(2v(t 1)),

    (8)v(t) = (t), 1 t 0.

    In Eq. (8), the parameter measures the frequency ofthe nonlinear function and will turn out to be an essentialparameter in the present study. Note that one can rescale andtranslate the variables such that Eq. (8) can be rewritten as


    dt= v + sin(v(t ) x0),

    (9)v(t) = (t), t 0.

    Equation (9) (also known as the Ikeda equation) was proposedby Ikeda et al. to model a passive optical bistable resonatorsystem and shows chaotic behaviors at particular parameterssuch as = 20,x0 = /4, and = 5 [19,21].

    In this paper, we will study the dynamical properties ofthe solutions of Eq. (8), both analytically and numerically. Wefocus in particular on the probabilistic properties of the chaoticsolutions. We then investigate chaotic solutions of{


    = vdvdt

    = v + sin(2v(t 1)), (10)v(t) = (t), 1 t 0,

    and characterize the statistical properties as completely as wecan. The main result is to show that Eq. (10) can reproduceexperimentally observed data of Brownian motion over a widerange of time scales, in spite of the fact that the evolutionequation is deterministic. Therefore, deterministic Brownianmotion can be generated from Eq. (10).

    The outline of the rest of this paper is as follows. Wefirst perform a bifurcation analysis for Eq. (8) in Sec. II. InSec. III we study the probabilistic properties of the chaoticsolutions numerically. In Sec. IV, we numerically examine thedynamics of the chaotic solutions of Eq. (10) and compare ourresults with recent experimental measurements of the mot